| Code |
14806
|
| Year |
3
|
| Semester |
S2
|
| ECTS Credits |
6
|
| Workload |
TP(60H)
|
| Scientific area |
Mathematics
|
|
Entry requirements |
Real Analysis and Differential Equations (Calculus I, II and III)
|
|
Learning outcomes |
On the completion of this course, students should be able of:
1. give an account of the foundations of calculus of variations and of its applications in mathematics and physics;
2. derive the Euler-Lagrange equations for variational problems,
3. solve variational problems with constraints: both algebraic and isoperimetric;
4. derive conserved quantities from symmetries, and use them to solve the Euler-Lagrange equations
5. analyse the local stability of the critical points of a variational problem.
|
|
Syllabus |
1. Motivation: brachistochrone, catenary, geodesics and minimal surfaces.
2. First variation and the Euler-Lagrange equation.
3. Isoperimetric problems.
4. Holonomic and non-holonomic constraints.
5. Newtonian, Lagrangian and Hamiltonian mechanics.
6. Noether theorem.
7. Second variation.
|
|
Main Bibliography |
1. The Calculus of Variations, Bruce van Brunt, New York: Springer, 2004.
2. Calculus of Variations: with applications to physics and engineering. Robert Weinstock. New York: Dover, 1974.
3. Métodos Matemáticos da Mecânica Clássica. V. I. Arnold, Moscovo: Mir, 1987.
|
|
Teaching Methodologies and Assessment Criteria |
1. The teaching methodology is based on theoretical-practical lessons. The theoretical part is based on the teacher's presentation of the syllabus contents, based on the bibliography of the unit or other notes available. In this introductory course to the calculus of variations, some technical proofs will be eventually omitted in order to get focus on the main ideas and applications. The practical part of the classes is based on solving exercises, both in an accompanying and autonomous way.
2. The assement is done through two written test (8+8 values) and exercises sheets for home resolutions and shorts presentations (4 values). The final classification will be given by by the sum of these evaluation elements.
|
|
Language |
Portuguese. Tutorial support is available in English.
|